Provided the flint etc hasn’t cut into the threads of the casing it needs to come out, and I then use superglue. It works for me, including on light tubular tyres.
If there’s any distortion to the tyre , such as a bulge or a ripple, I’d normally scrap the tyre. Occasionally a fabric patch inside the tyre and then deal with the cut will work. I’d probably only use the tyre on the rear though.